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No it is not posible, you are only able to have 1 Public Facing Website and that can only be assigned to one domain name. In theory the second domain will only be used for Email and Lync. The only way for you is to setup a website at your host for the second domain and depending on which plan you are make the relevent DNS changes.
P1 - You can edit the DNS records and add an A Record (example-Host=@ and IP=22.214.171.124) by getting the IP address of the site from your host. You would also want to add a CNAME record (example Host=www and address domainname.com). Adding these records would redirect web traffic for the domain.
E Plan - You would remove any forwardings for the web traffic of the 2nd domain at the host.
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This is Jason Burnside from Microsoft Office365 SharePoint Support.
Thanks for coming to the communities with your concerns about having two public websites.
Did you find Richard’s information helpful? Thanks for the post Richard.
Unfortunately, if you would like to have two public websites you would need to have a second subscription.
I will continue to monitor this thread if you have any additional questions.
Office 365 SharePoint Support
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